1 B-Trees Section 4.7

2 AVL (Adelson-Velskii and Landis) Trees AVL tree is binary search tree with balance condition –To ensure depth of the tree is O(log(N)) –And consequently, search complexity bound O(log(N)) Balance condition –For every node in tree, height of left and right subtree can differ by at most 1 How to maintain balance condition? –Rotate nodes if condition violated when inserting nodes –Assuming lazy deletion

3 B-Trees: Problem with Big `O’ notation Big ‘O’ assumes that all operations take equal time Suppose all data does not fit in memory Then some part of data may be stored on hard disk CPU speed is in billions of instructions per second –3GHz machines common now Equals roughly 3000 million instructions per seconds Typical disk speeds about 7,200 RPM –Roughly 120 disk accesses per second So accessing disk is incredibly expensive –We may be willing to do more computation to organize our data better and make fewer disk accesses

4 M-ary Trees Allows up to M children for each node –Instead of max two for binary trees A complete M-ary tree of N nodes has a depth of log M N Example of complete 5-ary tree of 31 nodes

5 M-ary search tree Similar to binary search tree, except that Each node has (M-1) keys to decide which of the M branches to follow. Larger M  smaller tree depth But how to make M-ary tree balanced?

6 B-Tree (or Balanced Trees) B-Tree is an M-ary search tree with the following balancing restrictions 1.Data items are stored at the leaves 2.Non-leaf nodes store up to M-1 keys to guide the searching Key i represents the smallest key in subtree ( i+1 ) 3.The root is either a leaf, or has between 2 to M children 4.All non-leaf nodes have between ceil(M/2) and M children 5.All leaves are at the same depth and have between ceil(L/2) and L data items, for some L.

7 B-Tree Example M=5, L=5

8 B-Tree: Inserting a value After insertion of 57. Simply rearrange data in the correct leaf.

9 B-Tree: Inserting a value (cont’d) Inserting 55. Splits a full leaf into two leaves.

10 B-Tree: Inserting a value (contd) Insertion of 40. Splits a leaf into two leaves. Also splits the parent node.

11 B-tree: Deletion of a value Deletion of 99 Causes combination of two leaves into one. Can recursively combine non-leaves

12 Questions How does the height of the tree grow? How does the height of the tree reduce? How else can we handle insertion, without splitting a node?